This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on their team. So far 17 people have turned up to these games. They are each represented by a different letter in the alphabet from A to Q. The players have different capabilities. The teams are usually 5 on 5 but it can vary if an odd number of people turn up for the game of if the captains agree that 2 weak players are equal to 1 strong player.
In the 7 rounds so far the scores have been:
ROUND1: TEAM 1 = AEHJK; TEAM 2 = BCDFIM; TEAM 1 SCORE = 10; TEAM 2 SCORE = 2; ROUND2: TEAM 1 = ABDFJ; TEAM 2 = CEGHI; TEAM 1 SCORE = 10; TEAM 2 SCORE = 3; ROUND3: TEAM 1 = ACEFK; TEAM 2 = BDGNO; TEAM 1 SCORE = 5; TEAM 2 SCORE = 1; ROUND4: TEAM 1 = ACFGI; TEAM 2 = BDJKPQ; TEAM 1 SCORE = 1; TEAM 2 SCORE = 4; ROUND5: TEAM 1 = ABIJ; TEAM 2 = CDFKOP; TEAM 1 SCORE = 3; TEAM 2 SCORE = 4; ROUND6: TEAM 1 = ABCFH; TEAM 2 = DEIKL; TEAM 1 SCORE = 10; TEAM 2 SCORE = 1; ROUND7: TEAM 1 = ACDIJ; TEAM 2 = BEFHK; TEAM 1 SCORE = 9; TEAM 2 SCORE = 0;
The organizers want to understand the strengths of each of the players so that they can organize more even match ups. If a teams strength is the sum of the strengths of each of the players then what is the strength of each player? It's possible that some combinations of players lead to a better result than would be predicted by a simple additive model. Is there a better way to accurately predict the results of the teams combinations? What is it?